## Frequency simulation 

Model is based on https://www.dsprelated.com/showarticle/973.php

### Time domain block diagram 

![time domain](freqsim_timedomain.png)

### Z domain block diagram

![Z domain](freqsim_zdomain.png)

## DDMTD Phase detector gain ($K_{PD}$)

$$
\begin{align*}

\Delta tag[n] &= \dfrac{N}{2\pi} \cdot \Delta\theta[n] \\[10pt]
\Delta tag(z) &= \dfrac{N}{2\pi} \cdot \Delta\theta(z) \\[10pt]
\dfrac{\Delta  tag(z)}{\Delta \theta(z)} &= \dfrac{N}{2\pi}   \\[10pt]
K_{PD} &= \dfrac{N}{2\pi}   \\[10pt]

\end{align*}
$$

## Loop Filter 

Low pass biquad filter calculator: https://www.earlevel.com/main/2021/09/02/biquad-calculator-v3/

## Si549 DCXO gain ($K_{DCXO}$)

Consider Si549 as a phase accumulator:
$$

\begin{align*}
\theta_{DCXO}[n] &= 2\pi f[n] \cdot T + \theta_{DCXO}[n-1] \\[10pt]
\theta_{DCXO}[n] &= 2\pi \bigg( f_{center} + 0.0001164 \cdot (\text{BASEADPLL} + ADPLL[n]) \cdot \dfrac{f_{center}}{10^6} \bigg) \cdot T_s + \theta_{DCXO}[n-1]  \\[10pt]

\theta_{DCXO}[n] &= \underbrace{ 2\pi \bigg(0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot T_s \bigg)}_{K_{DCXO}} \cdot ADPLL[n]  +
 \underbrace{ 2\pi \bigg( f_{center} + 0.0001164 \cdot \text{BASEADPLL} \cdot \dfrac{f_{center}}{10^6} \bigg) }_{f} \cdot T_s + \theta_{DCXO}[n-1]  \\[20pt]

K_{DCXO} &=  2\pi 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot T_s  \\[10pt]
K_{DCXO} &=  2\pi * 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot \dfrac{1}{f_s}  \\[10pt]
K_{DCXO} &=  2\pi * 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot \dfrac{N}{f_{center}}  \\[10pt]
K_{DCXO} &= \dfrac{0.0001164\cdot N\cdot 2\pi}{10^6} \\[10pt]


\end{align*}
$$